And like I said, ridiculous question as what point the "real" unless you think there is some hidden or unique meaning to the numbers other than just being the product of the need to count efficiently and that we are symbol users?

That's better as I'm not attaching a mysterious meaning to them, but they do have interesting ways of behaving, almost like they have a life of their own.

Philosophy Explorer wrote:Not true. There was no "formal axiomatic system" at the time the natural numbers came about in history. Does Wiki support you on this? Which article if so?

There were no 'natural numbers' at the time counting came about and your 'natural numbers' are the product of Arabic numerology and that the most common base numeral system is ten - not surprising given we have ten fingers. Axiomatic systems are one of the ways Mathematics invented to describe and understand the natural numbers and their operations within Mathematics, as was Set Theory. I'd be loathe to use Wiki as the be all and end all of such things but I'd be bloody surprised if the search 'Peano Arithmetic' and 'Axiomatic systems' didn't provide back-up for what I say.

Philosophy Explorer wrote:Not true. There was no "formal axiomatic system" at the time the natural numbers came about in history. Does Wiki support you on this? Which article if so?

There were no 'natural numbers' at the time counting came about and your 'natural numbers' are the product of Arabic numerology and that the most common base numeral system is ten - not surprising given we have ten fingers. Axiomatic systems are one of the ways Mathematics invented to describe and understand the natural numbers and their operations within Mathematics, as was Set Theory. I'd be loathe to use Wiki as the be all and end all of such things but I'd be bloody surprised if the search 'Peano Arithmetic' and 'Axiomatic systems' didn't provide back-up for what I say.

So you're playing that game now. There was no Arising_uk either at the time you were born.

The Arabs did not invent Arabic numerals, the Hindus did, and the Arabs got the idea from the Hindus. Europeans got the "Arabic numerals" from the Arabs, and assumed that the Arabs invented the system, not knowing that the Arabs borrowed from the Hindus.

You can count without any number system by matching. A quick example is when a person sees that every seat in an arena is filled by a person sitting in each seat, regardless of what the actual number of seats or people in the arena is, we know that the number is the same. So, one can "count" in this manner without using natural numbers, but without being able to match two different sets of objects together, counting would have been done by the use of natural numbers. However, my point was, and remains, that natural numbers have limitations, which is why other numbers were developed, the last being complex numbers. In fact, all numbers can be thought of as complex numbers in disguise, and we even have a theorem that tells us there will be no need to develop any additional numbers beyond complex numbers. Natural numbers are useful for addition and multiplication, where we have closure --- add or multiply any two natural numbers together and you get a natural number, so one remains in the set of natural numbers when using the operations of addition and multiplication. However, this closure property fails when doing simple subtraction. Then, we need to develop integers, adding zero and negative numbers, in order to have closure. But, integers fail when we add in the operation of division. Then we have to add in rational numbers. But, there are more irrational numbers than rational, so we have to add in real numbers. Then, we have difficulty finding roots, so we add in complex numbers.

I think Phil X's general point is an interesting one, however, concerning the nature of numbers. I'll expand on his point from natural numbers to all numbers, including complex numbers. Complex numbers give us some unexpected surprises. For example, an exponential function grows and grows at an accelerating pace, anyone can see this when graphing e raised to the x power, and just plug in natural numbers for x, one will see that curve's graph take off towards infinity. We can also graph sin(x) and cos(x) and see that we get a graph that bounces back and forth between 1 and minus 1. So, whoever would have thought that an exponential function would equal a function consisting of a sin and cos function? I sure as hell wouldn't; however, they do equal each other when we plug in complex numbers. I think this is an illustration of the point Phil X is making ---- that we cannot just manipulate the numbers we come up with in any old manner, the logic surrounding them contains a life of their own.

As far as the metaphysical question, I'm not convinced that the only two options are a realism based on some form of Platonism versus purely made--up fiction from a human mind. I think rather it's a combination. Until the numbers are invented, by the human mind, I fail to see how they could exist, because they do not exist outside of us in the universe outside our minds. However, once invented, then they do have a logical structure, and do have an existence that is at least somewhat independent from us, as we cannot merely manipulate the numbers any way we please.

The thing is the issue being raised here cannot be answered within mathematics, because it is a philosophical question, which is why even top mathematicians disagree with each other on this issue.

Science Fan wrote:The Arabs did not invent Arabic numerals, the Hindus did, and the Arabs got the idea from the Hindus. Europeans got the "Arabic numerals" from the Arabs, and assumed that the Arabs invented the system, not knowing that the Arabs borrowed from the Hindus. ...

Well for sure but the ones we use now are from the Arabs interpretations.

You can count without any number system by matching. A quick example is when a person sees that every seat in an arena is filled by a person sitting in each seat, regardless of what the actual number of seats or people in the arena is, we know that the number is the same. So, one can "count" in this manner without using natural numbers, but without being able to match two different sets of objects together, counting would have been done by the use of natural numbers. ...

I'd have thought that more accurately it would be before counting was done with the number symbols we'd have used 'pebbles' or the scratch.

I think Phil X's general point is an interesting one, however, concerning the nature of numbers. I'll expand on his point from natural numbers to all numbers, including complex numbers. Complex numbers give us some unexpected surprises. For example, an exponential function grows and grows at an accelerating pace, anyone can see this when graphing e raised to the x power, and just plug in natural numbers for x, one will see that curve's graph take off towards infinity. We can also graph sin(x) and cos(x) and see that we get a graph that bounces back and forth between 1 and minus 1. So, whoever would have thought that an exponential function would equal a function consisting of a sin and cos function? I sure as hell wouldn't; however, they do equal each other when we plug in complex numbers. I think this is an illustration of the point Phil X is making ---- that we cannot just manipulate the numbers we come up with in any old manner, the logic surrounding them contains a life of their own. ...

My point I'd have thought? As the logic surrounding them will be dependent upon the operators, functions and proof methods in use, nothing mysterious about it. That one can't see all the possible paths would be due to our calculating inadequacies I'd have thought hence the interest in automated theorem provers. Although my Logic is not strong enough to know about possible problems with this approach with some of the axiom systems in mathematics.

As far as the metaphysical question, I'm not convinced that the only two options are a realism based on some form of Platonism versus purely made--up fiction from a human mind. I think rather it's a combination. Until the numbers are invented, by the human mind, I fail to see how they could exist, because they do not exist outside of us in the universe outside our minds. However, once invented, then they do have a logical structure, and do have an existence that is at least somewhat independent from us, as we cannot merely manipulate the numbers any way we please. ...

But that structure is just the structure of the relations created by the axioms and the symbols surely?

The thing is the issue being raised here cannot be answered within mathematics, because it is a philosophical question, which is why even top mathematicians disagree with each other on this issue.

If it were simply a matter of structures, then we could devise any systems we desire, but we cannot do so. That was actually the goal of mathematicians at one time, until it was proven that the goal will never be reached.

There are many Platonists in mathematics. Almost all young mathematicians start off as Platonists. Roger Penrose, as one example, incorporates a Platonic view into his metaphysics.

We can invent a triangle, but once invented, the properties of the triangle are not things we make-up, or invent, but discover.

The thing is it's not entirely clear that the inventions of mathematics have nothing to do with the empirical world. Geometry, trigonometry, probability and statistics, certainly seem to be derived from dealing with the real world. In some ways, math is pure logic, and in others, it appears more like a physical science.

Philosophy Explorer wrote:You shouldn't make an assumption like that about me as the body of knowledge concerning natural numbers keeps growing. ...

Then why did you say "What is the real meaning..."?

It started off simply and has evolved over the years with discoveries that keep going on. The most basic question about natural numbers is whether they really exist or just in our minds? ...

What on earth do you mean by "really exist" with respect to numbers? At base the natural numbers were created as symbols for counting objects and all they really are is shorthand for '1' and lots of '1''s.

Let me ask you this. What do natural numbers mean to you? ...

They mean that I can count or describe the number of things easier.

Do you think they're the most important type of number or do you have a greater interest in other types of numbers? ...

How do even rank numbers by 'importance'?

Do you even know what a natural number is? ...

Sure, the integers.

What hidden properties do natural numbers have that await discovery?

I doubt they have any hidden properties as they are just shorthand for lots of '1's. Are there relationships between the symbols due to the existence of the mathematical operators that we haven't found yet in the axiomatic system for the integers, I guess so but you'd have to ask a Mathematician as Logic is just about my game not Maths.

I will answer in reverse order.

"I doubt they have any hidden properties as they are just shorthand for lots of '1's. Are there relationships between the symbols due to the existence of the mathematical operators that we haven't found yet in the axiomatic system for the integers, I guess so but you'd have to ask a Mathematician as Logic is just about my game not Maths."

I could ask what you're doing on this thread since logic is your game but not math, but since you decided to intrude, I'll deal with it. I presume by"...just shorthand for lots of '1's", you mean something like 4 = 1 + 1 + 1 + 1 e.g. How about 2 x 2 = 4? Not quite the shorthand you envisioned,
is it? Then you said "...axiomatic system for the integers." This "system" isn't taught in college (there is an axiomatic system for the real numbers which is a different story) so you won't find any mathematical operators there and logic which you said is your game won't help you out here either.

Do you even know what a natural number is? ... "Sure, the integers." Really!!!? What does your logic say about -1? That's an integer, but not a natural number. I suggest you refresh your memory about the difference between integers and natural numbers by looking up their definitions because I know you "just" won't take my word for it.

"How do [we] even rank numbers by 'importance'?" Well, for one thing, different types of numbers arose throughout history which allowed mathematicians and scientists to make progress in many different ways in their investigations of phenomena. Also importance can be a personal preference. For example an accountant would look at numbers differently than a Sudoku puzzle lover. For you, you said you can count or describe the number of things easier.

"What on earth do you mean by 'really exist' with respect to numbers? At base the natural numbers were created as symbols for counting objects..." As I had asked, do they just exist inside of your mind or do they have physical existence?

"Then why did you say 'What is the real meaning...'?"

As more discoveries are made about how the natural numbers interact among themselves, the more interesting they get as they show unusual properties and discoveries are made all the time about them in numerous ways. Under these circumstances, it would be natural to ask what is the real meaning of natural numbers.

PhilX

Multipication can be observed as addition adding itself (self-reflecting)